Page 356 - Design and Operation of Heat Exchangers and their Networks

342 Design and operation of heat exchangers and their networks

For the inverse Laplace transform to the time domain, numerical inver-
sion is also a reasonable choice. Among the various numerical inversion
methods, the Gaver-Stehfest algorithm (Stehfest, 1970; Jacquot et al.,
1983) and the algorithm based on Fourier series (Ichikawa and Kishima,
1972; Crump, 1976) are often applied. Let f(z) and F(s) be a Laplace-
transform pair; these two algorithms are represented as follows, respectively:
(1) Gaver-Stehfest algorithm
N
ln 2 X ln 2
fzðÞ ¼ K n Fn ð even NÞ (7.95)
z z
n¼1
where the constant integer N is suggested to be 10 for 8-digit arith-
metic and 18 for 16-digit arithmetic, and the coefficient K n is given by

min n, N=2ð Þ N=2
2k
X k ðÞ!
n + N=2
K n ¼ 1ð Þ (7.96)
ð N=2 kÞ!k! k 1Þ! n kÞ! 2k nÞ!
ð
ð
ð
k¼ n +1Þ=2
ð
This algorithm cannot be used if there are oscillatory components or
very rapid changes in f(z)(z>0).
(2) FFT algorithm
" #
M 1
e aτ n X 2iπnk=M 1
ð
f τ n ¼ Re fa + ikπ=τÞe faðÞ ð 2:178Þ, (7.97)
e
ðÞ
e
τ 2
k¼0
in which z n ¼nz/M (n M/2), a is a constant, and 4<az<5, and
the series
M 1
X ikπ 2π nk
f n ¼ Fa + exp i (7.98)
z M
k¼0
is calculated by means of the fast Fourier transform algorithm (FFT).
If the original function f(z) has discontinuity points, one should pay
attention to the additional oscillation (Gibbs phenomenon) near these
points.
7.2.3 Numerical methods
In most cases, analytical solutions are difficult to obtain, while numerical
solutions are always available. The three models earlier describing the heat
exchanger dynamic behaviors, namely, lumped parameter model, distrib-
uted parameter model, and cell model, can all be solved with numerical
methods. Among all numerical methods, the finite-difference technique